Distributions defined by q-supernomials, fusion products, and Demazure modules

We prove asymptotic normality of the distributions defined by q-supernomials, which implies asymptotic normality of the distributions given by the central string functions and the basic specialization of fusion modules of the current algebra of sl2. The limit is taken over linearly scaled fusion powers of a fixed collection of irreducible representations. This includes as special instances all Demazure modules of the affine Kac-Moody algebra associated to sl2. Along with an available complementary result on the asymptotic normality of the basic specialization of graded tensors of the type A standard representation, our result is a central limit theorem for a serious class of graded tensors. It therefore serves as an indication towards universal behavior: The central string functions and the basic specialization of fusion and, in particular, Demazure modules behave asymptotically normal, as the number of fusions scale linearly in an asymptotic parameter, N say.


Introduction
The q-supernomials coefficients are q-analogues of the coefficients of the monomials x a in the expansion of m i=1 (1 + x + . . . + x j ) L j . They have a combinatorial interpretation in terms of generalized Durfee dissection partitions in that they are generating functions of so-called (L 1 , . . . , L m )admissible partitions with exactly a parts [35, §3]. In there most simple form, when m = 1, they are the well-known q-binomial coefficients which are the generating functions for restricted partitions [2, §3], a result that goes back to Gauss and coins them Gaussian binomial coefficients. Let us mention that, geometrically, the coefficients of q-binomials count the number of invariants of binary forms [39, §3.3], [1,Corollary].
The main focus on q-supernomials, from our point of view, lies on their appearance as characters of finite-dimensional modules in the representation theory of infinite-dimensional Lie algebras. Precisely, they describe the string functions in fusion modules, i.e. graded tensor products, of the current algebra sl 2 ⊗ C[t] [12,13]. Consequently, their coefficients encode the dimensions of weight spaces, i.e. isotropic components with respect to the action of a maximal abelian subalgebra.
The exact determination of those coefficients is certainly possible in any fixed instance of a q-supernomial, simply by extrapolation from the definition of q-supernomials and their building blocks, the q-binomials. In general though, the explicit description of those coefficients remains intractable and one usually is satisfied with concrete expressions for their generating function, the q-supernomial. We will examine those coefficients from a qualitative point of view by the interpretation of q-supernomials as probability generating functions of discrete distributions. Furthermore, we investigate the generating function of the q-supernomials themselves. This generating function equals the so-called basic specialization of the character of sl 2 ⊗ C[t]-fusion modules, and can be understood as the Hilbert polynomial associated to the parts that are graded by the action of t. Our main results are Theorem 4.20 and Theorem 4. 24, and can be summarized as follows: Along with complementary results for the Demazure module V −N ω 1 (Λ 0 ) of the higher rank affine Kac-Moody algebra sl r [5, §5.4], our findings highlight the first serious indication of a central limit theorem concerning the central string functions and the basic specialization of fusion modules. We will conclude this line of thought by a discussion of fusion of symmetric power representations in type A and the following conjecture.  Then, the central string functions and the basic specialization of F µ (N ) behaves asymptotically normal as N → ∞.
Since Demazure modules of sl 2 are special instances of fusion modules we will compare (see §4.5.2) our results to previously derived formulae for the expectation value of the basic specialization of Demazure modules [7]. There the expectation value has been derived via different methods, through the detailed analysis of the recursion given by Demazure's character formula.
Our results are accompanied by conjectured local central limit theorems in §5, that give the first hint towards the asymptotic growth of dominating weight multiplicities in fusion modules found in the literature. We furthermore discuss the geometric interpretation in terms of rational points of unipotent partial flag manifolds over finite fields in Remark 6.4.

Notation
2.1. q-supernomial coefficients. We follow [35] and fix a vector L = (L 1 , . . . , L m ) ∈ Z m + , and let l m = m i=1 iL i . Consider the expansion [35, (2.5 and the fermionic representation [35, (2.8)] of the coefficient as a convex sum of products of usual binomial coefficients. Such an expression as a positive sum of products of binomials is commonly referred to as fermionic. Schilling and Warnaar define the q-supernomials [35, (2.9)] as a q-analogue of those coefficients Here j m+1 = 0, M k q = [M ]q!
[k]q![M −k]q! denotes the q-binomial coefficient and [k] q ! = k i=1 1−q i 1−q the q-factorial (see e.g. [10]). Note that Feigin and Feigin describe in [12,Theorem 5.1] the characters of sl 2 ⊗ C[t] fusion modules in terms of a slight modification of L a q , that is [35, (3.1)] T (L, a)(q) = q l 1 a L a − lm 2 1 q , with its explicit form [35, (3.2)] T (L, a)(q) = Again, we let j m+1 = 0. We will discuss characters of fusion modules as generating functions of q-supernomials in greater detail later in §3. This explicit expression ofT can be derived easily from the above definition of L a q due to the symmetry of the q-binomial coefficients For illustration purposes, in the case m = 1 the definitions (2.1), (2.2) above give That is,T is a shifted (here by L 1 /2) and translated (here by a 2 ) version of (L 1 ) a q . 2.2. Lie algebras, Demazure and fusion modules. For general facts about current and affine Kac-Moody algebras and their representation theory we refer the reader to [8,23]. We denote by sl r the complex-valued r × r matrices with trace 0 (the reader might adapt to the case r = 2 as this will be the setting we will mostly consider). Then, sl r ⊗ C[t] denotes its current algebra and, closely related, sl r its associated affine Kac-Moody algebra which can be realized as the extended loop algebra of sl r , i.e. sl r = sl r ⊗ C[t, t −1 ] ⊕ Cd ⊕ Cc (see [23, §7]). Here, d denotes the derivation t d dt and c the canonical central element. In view of that realization letb ⊃ĥ be the Borel and Cartan subalgebra in sl r corresponding to their a priori fixed counterparts in sl r . We denote the simple roots by α 0 , α 1 , . . . , α r−1 ∈ĥ * , the highest root by θ = α 1 + . . . + α r−1 , the imaginary root by δ = α 0 + θ, and the simple coroots by α ∨ 0 , . . . , α ∨ r−1 ∈ĥ. Note that we assume α 1 , . . . , α r−1 to correspond to the standard embedding of sl r . Let s 0 , s 1 , . . . , s r−1 be the simple reflections associated to the simple roots and let the subgroup W aff = s 0 , s 1 , . . . , s r−1 of GL(h * ) denote the affine Weyl group.
One usually writes the monomials in the characters of such modules as e µ , the coefficient k in the monomial e −kα 0 is referred to as the degree. The Λ 0 , Λ 1 , . . . , Λ r−1 are called fundamental weights, the V (Λ l ) the fundamental representations and V (Λ 0 ) the basic representation.
We will not be directly concerned with the representation theory of sl r but instead with Demazure and fusion modules, i.e. finite-dimensional representations of the Borel subalgebrab and the current algebra sl r ⊗ C[t], respectively (see e.g. [9,12,13,14,19,24]).
For w ∈ W aff and a dominant integral weight Λ = m 1 Λ 0 + m 2 Λ 1 + . . . + m r−1 Λ r−1 denote the associated Demazure module by V w (Λ). Such a Demazure module is defined as the finite-dimensional subspace of the integrable highest weight representation V (Λ) generated by the action of the universal enveloping algebra U(b) on a non-zero weight vector Demazure's character formula [11,28,31] allows the computation of the character χ(V w (Λ)) by an iterated application of certain operators on the monomial e Λ = χ(V 1 (Λ)), as follows. We introduce the convention that Note that this is natural in the sense that Gauss's summation formula k i=0 i = k(k+1) 2 extends to all k ∈ Z, as does the identity k i=0 1 = k + 1. With this convention the Demazure operator D j associated with a simple reflection s j acts on monomials e µ as For an arbitrary Weyl group element w ∈ W aff we choose a reduced decomposition w = s j 1 s j 2 · · · s j l and set D w = D j 1 D j 2 · · · D j l . Demazure's character formula now states that the character of V w (Λ) can be computed recursively as For a general definition of a fusion module consider irreducible sl r -modules V (λ 1 ), . . . , V (λ n ) with integral dominant weights λ = (λ 1 , . . . , λ n ). For regular Z = (z 1 , · · · , z n ) ∈ C n let sl r ⊗ C[t] act by The associated graded module is called the fusion module: The additional grading is induced by the powers of t. We obtain a graded tensor product multiplicity m q µ ∈ Z[q] such that m µ = m q µ (1), where m µ is the usual tensor product multiplicity of V (µ) in ⊗ i V (λ i ). For specific λ = (λ 1 , . . . , λ n ) it is known that the fusion modules F λ (Z) are independent of the choice of the evaluation parameter Z, see e.g. [ [3,4,15,16]. All our random variables X will be discrete and finite. Recall that the expected value of such a random variable is the weighted average E(X) = x P(X = x)x. The covariance of two random variables X and Y is Cov(X, Y ) = E((X − E(X))(Y − E(Y ))). They are said to be uncorrelated if Cov(X, Y ) = 0. The variance of X is Var(X) = Cov(X, X). Its probability generating function is E(q X ) = x P(X = x)q x , and the associated probability distribution µ X = x P(X = x)δ x . Here, δ x denotes the Dirac distribution (point mass) at x. A sequence X N converges P-almost surely (P − a.s. for short) to X if P(lim N →∞ X N = X) = 1. Convergence and equality in distribution will be denoted by d −→ and d =, respectively. N (µ, Σ) will denote the normal distribution with mean µ and covariance matrix Σ. Note that N (µ, 0) = δ µ . The conditional probability P(Y = y|X = x) = P(X = x) −1 P(X = x, Y = y) is the probability of Y taking the value y given the occurence of the value x for X. A mixture of distributions µ X i is a finite convex combination thereof, i.e. i w i µ X i for some weights w i ≥ 0 with i w i = 1. The probability generating function of a mixture is i w i E(q X i ).

Characters and their specializations
3.1. Graded characters of fusion and Demazure modules of sl 2 . We focus on [12,13] which study in detail fusion modules for the current algebra sl 2 ⊗ C[t]. Note that irreducible representations of sl 2 are indexed by natural numbers N 0 = {0, 1, 2, 3, . . .}. Hence, a fusion module F d (Z) for sl 2 is given by a dimension vector d = (d 1 , . . . , d n ) ∈ N n 0 and regular Z ∈ C n . For sl 2 it is known that the fusion modules are independent of the choice of Z ∈ C n [12], and it is usual to omit the evaluation parameter in the notation. In particular, Feigin and Feigin [12] study the fusion module F d with dimension vector d = (1 b 1 , 2 b 2 , . . . , (m + 1) b m+1 ), and denote it They prove [12, Theorem 5.1] that, written here in terms of the q-supernomial T (L, a)(q) (2.2), its graded character can be described as It is well-known that Demazure modules V w (Λ) associated to sl 2 carry a sl 2 ⊗C[t]-module structure and as such are special instances of fusion modules (see e.g. [9, §1.5.1] or [19, §3.5]). To be precise, we have isomorphisms of sl 2 ⊗ C[t]-modules as follows Here, we write l(w) for the length of a reduced decomposition of w ∈ W aff . All elements of W aff have the form Then, one can precisely identify the characters of Demazure modules via those of fusion modules as follows.
Proposition 3.1. Consider the Demazure module V w = V w (mΛ 0 + nΛ 1 ) of sl 2 of (fixed) highest weight Λ = mΛ 0 + nΛ 1 and recall the character formula for fusion modules (3.1). The character χ(V w )(z, q), written in the coordinates z = e −α 1 , q = e −δ , is given by For the trivial element w = 1 one has χ(V 1 )(z, q) = e Λ . For w = (s 0 s 1 ) N , N ≥ 0 one has (3.5) For w = s 0 (s 1 s 0 ) N −1 , N > 0 one has (3.7) The sum of the entries in L w represents the length l(w) of the Weyl group element w. When either n or m equals 0, then L w = (0, l(w)).
Proof. Feigin [13, (11)] denotes an integrable highest weight representation and d.v i,k = 0. In our notation, the canonical central element is c = α ∨ 0 + α ∨ 1 , the coroot is h 0 = α ∨ 0 , and the scaling element d is given by α 0 (d) = 1 and α 1 (d) = 0. Therefore, by comparison of the highest weight vector we have L i,k = V ((k − i)Λ 0 + iΛ 1 ). The bigrading is chosen according to the action of h 0 and d, and consequently, the character is denoted in the monomials e α 0 = e δ−α 1 and e −δ , respectively. By [13, Corollary 3.1] each such module L i,k can be constructed as an inductive limit of fusion products, that is L i,k = C i+1 * (C k+1 ) 2∞ . Each fusion product can be identified with the corresponding Demazure module V w ((k − i)Λ 0 + iΛ 1 ) by comparing the weights of the extremal weight vectors described in [13, §1]. Now apply the character formula [12, Theorem 5.1], noting that e α 0 = zq −1 .
Note the explicit form (compare (2.2)) and see Figure 2 for the actual plot. (3) The character of the Demazure module V (s 1 s 0 ) 2 (2Λ 0 ) written as a Laurent polynomial in the monomials z = e −α 1 , q = e −δ , i.e. the generating function of its weight space dimensions, This generating function can be compute via Demazure's (recursive) character formula Figure 3 for the actual plot. As to the relations between the different generating functions, one has All expressions are equivalent up to translations (multiplication by e.g. z −4 ), reflections (evaluation at the reciprocal 1/q), and rotations (evaluation at mixed monomials zq −4 ), respectively. Note that the example (3.9) is an instance of the general formula (3.6).

3.2.
Real characters and the basic specialization. As already mentioned in §2.2, the specialization of the character of irreducible representations of sl 2 . Note that frequently this type of specialization is called a real character [33]. Therefore, (3.2) implies This so-called factorization phenomenon for Demazure modules is known independent of the theory of fusion modules, first proved in [32,33], and generalized considerably, e.g. [18]. Note, since characters of (non-graded) tensor products multiply, i.e. χ(V ⊗ W ) = χ(V ) · χ(W ), the associated distributions convolute and asymptotic considerations, as l(w) → ∞, take the simplest form of a central limit theorem for sums of i.i.d. random variables. Therefore, characters of usual tensor products, and equivalently the above specializations at q = 1, are well understood from a statistical point of view and have been analyzed further in great detail, e.g. [42].
Much less studied is the so-called basic specialization of those characters, i.e. their evaluation at z = 1. We borrow this terminology from Kac [23, §1.5, 10.8, 12.2] who analyzed this kind of specialization for characters of integrable highest weight modules V (Λ), and obtained Macdonald's identities for Dedekind's η-function [23, §12.2]. Note that the evaluation of Demazure characters at z = 1 can be equivalently described as their restriction to the subspace Cd when viewed as functions onb . For specific Demazure modules the basic specialization has a geometric interpretation in terms of Galois numbers. That is, consider the translations t −N ω 1 = (s 1 s 2 . . . s r−1 σ r−1 ) N in the extended affine Weyl group of sl r , where ω 1 = Λ 1 −Λ 0 and σ denotes the automorphism of the Dynkin diagram of sl r which sends 0 to 1. We denote the Demazure module associated to the fundamental weight Λ 0 and those translations by equals the so-called generalized Galois number that counts the number of flags 0 = V 0 ⊆ V 1 ⊆ · · · ⊆ V r = F N q of length r in an N -dimensional vector space over a field with q elements [5,27,43]. For r = 2, that is in the case of Demazure modules of sl 2 , this is the Galois number that counts the number of subspaces in finite vector spaces that has been studied initially by Goldman and Rota [20]. In general, due to a result of Shimomura [37], q-supernomials count the number of points of unipotent partial flag varieties defined over finite fields (see §6 and, in particular, Remark 6.4 therein).

Distributions defined by q-supernomials
Let L := (L 1 , . . . , L m ) ∈ Z m + and consider the q-supernomial given in Recall that these generating functions were introduced by Schilling and Warnaar [35] who showed that they enumerate L-admissible partitions with exactly a parts. In this section we study the average behavior of the distributions defined byT (L, a)(q) and by the generating functioñ of the total number of L-admissible partitions that equals the specialization χ(F(0, L))(1, q) by (3.1). We show that the total number and in certain (typical) cases the a-restricted number of L-admissible partitions are asymptotically normally distributed with asymptotic parameter being a convergent sequence 1 N L (N ) , as N → ∞. 4.1. Preliminaries. The distributions with probability generating function F a,b (q) := a+b a q / a+b a were first investigated by Mann and Whitney [30], who showed: as a, b → ∞.
A corresponding local limit theorem was proved by Takacs [41].
Remark 4.2. The q-binomials enumerate different number theoretic [2], geometric [39], and combinatorial [41] objects. We interpret them here as counting inversions [40]. Consider a word (unordered sequence) w = (w 1 , . . . , w n ) of elements from an ordered set. A 4-tuple (i, j, w i , w j ) with i < j and w i > w j is called inversion. It is well known that a+b a q is the generating function for inversions in words of a zeroes and b ones.
We call a vector j = (j 1 , . . . , . . m, and a probability distribution L-compatible if the set of L-compatible values has probability one. For L-compatible j we let Inv(L, j) denote a random variable with probability generating function here and in the sequel j m+1 = 0), and we let We view the normalized fermionic expression g(q) =T (L)(q)/T (L)(1), and g a (q) =T (L, a)(q)/T (L, a)(1), respectively, as mixtures of probability generating functions, i.e.
All generating functions and random variables considered here depend implicitly on the admission vector L, but this dependence will from now on for convenience be suppressed in the notation. We may equivalently write g(q) as follows.
Proof. By the definition of the conditional distribution Our interest lies in the distribution of the random variable is (by the properties of conditional expection) uncorrelated to (any square-integrable function of) J. Furthermore, by Theorem 4.1 we have Let us call the distribution of J the mixing distribution (for the total number of L-admissible partitions). To understand the behavior of mixing distributions and to introduce a convenient way to refer to them, let us describe a simple random experiment. We focus our attention on the unrestricted case first.
) denote the random variable the number of appearances of letter i in word W (k). Then, the ran- Clearly, under the assumptions above, S L is the sum of independent uniformly distributed random variables and we have . We may thus interpret the supernomial coefficients as the probabilities of the random variable S L : Schilling and Warnaar [35, (2.8)] give the representation and the following result.
Lemma 4.4. For j = (j 1 , . . . , j m ) ∈ N m 0 (and with the usual convention about binomial coefficients that a b = 0 unless 0 ≤ b ≤ a) let Then, these numbers define a L-compatible probability distribution.
Proof. Indeed, we have a probability distribution since The L-compatibility is obvious.
We may describe this probability distribution alternatively as follows.
Then, the joint distribution of (J 1 , . . . , J m ) is as in Lemma 4.4.
Proof. We use (formal) generating functions. It is clear that the joint generating function of (J 1 , . . . , J m ) as defined in (4.4) is Now extract coefficients to see that this corresponds to the distribution defined in Lemma 4.4 Remark 4.6. Since A k−i+1 (k) = B k−i+1 (k)+B k−i+2 (k)+· · ·+B k (k) counts the number of appearances of the highest i letters in word W (k), the J i may be described as the total number (overall count) of the i highest non-zero letters in all words.
Proposition 4.5 shows that the mixing distribution for the total number of L-admissible partitions may be realized as a simple linear transformation of B L . We call B L the underlying occupancy distribution. This representation can be used for explicit calculations, and reduces the asymptotic treatment of J in the unrestricted case to the well known asymptotics of multinomial distributions. Let us recall the following classical result about the asymptotic normality of multinomial distributions.
Theorem 4.7. Let B (N ) have the multinomial distribution with parameters N and p = (p 0 , p 1 , . . . , p m ). Then, we have mean E(B (N ) ) = N p, covariance matrix Cov(B (N ) ) = N Σ, and Given these initial observations, we have a straightforward program to treat the asymptotics of the random variable T = Q(L, J) + Y : (1) split T into an "occupancy part" Q(L, J)+E(Y |J), dependent only on J and a remaining "rest-inversion part" R = Y −E(Y |J) "orthogonal" to J, (2) find the asymptotics of the two parts, using Theorem 4.1 and Theorem 4.7, (3) combine the results. This program is carried through in §4.2.

4.1.2.
The probabilistic setup for the a-restricted case. It is clear that the same experiment describes the a-restricted case when we consider only the outcomes with total sum S L = a. That is, for the a-restricted case the underlying occupancy distribution is B L | S L = a, i.e. the distribution of B L , conditioned to have S L = a. In order to have a succinct wording for the "most important" restricted cases we make the following definition (Cf. (4.2)). We first show a result that enables us to treat the occurring quadratic functions of J (N ) . For quadratic functions of asymptotically normal random vectors X (N ) one has in general: Proposition 4.9. Assume there exists b ∈ R m and a positive semidefinite matrix Σ ∈ R m×m of positive rank such that Let M ∈ R m×m , v ∈ R m , and consider the quadratic function q( where For the convergence of moments we have here:

Now, clearly A (N )
Proposition 4.10. In the situation of Proposition 4.9 assume that addi- and, If furthermore E X We omit the elementary proof, and now look at the asymptotic behavior of the conditional distribution of R (N ) given that J (N ) = j (N ) . where Thus under the conditions above 1 v(a, b)). Theorem 4.13. Let R(a, b) denote a random variable with distribution N (0, v(a, b)). If 1 N L (N ) → a = 0, and if there exists b ∈ R + m and a positive semidefinite matrix Σ ∈ R m×m of positive rank such that Then, as N −→ ∞, where the constituents on the right-hand side are independent.
Proof. Let A ⊂ R be a Borel set and f : R m −→ R be bounded and continuous. We have
Therefore, v(a, b)) A E f (X) .
Let us collect some immediate corollaries.
Corollary 4.14. In the situation of Theorem 4.13 assume that additionally ), and c = c(a, b) be the vector with coordinates Proof. The first assertion follows directly from Proposition 4.9. For the second assertion observe that by Theorem 4.13 the limiting distribution is the convolution of the normal distributions R(a, b) and the limiting distribution in (4.5).
Corollary 4.15. In the situation of Theorem 4.13 assume that additionally Finally, we note for the convergence of the variance of R (N ) : Lemma 4.16. In the situation of Theorem 4.13 assume that additionally and by our assumptions E (R (N ) ) 2 |J (N ) /N 3 converges boundedly to v(a, b).

4.3.
Unrestricted number of parts. We first consider the total number T (L)(q) of L-admissible partitions, without restrictions on the number of parts. In this case clearly A k−i+1 (k) (as defined in Proposition 4.5) has a binomial distribution with parameters n = L k and p = i k+1 , and hence each J i can be represented as a sum of independent binomial variables. Furthermore, the covariance of A k−i+1 (k) and A k−j+1 (k) can be computed as We therefore have Moreover, straightforward computations lead to the exact expectation value of the random variable defined in (4.1). We will need this for comparison to the basic specialization of Demazure modules in §4.5.2. Note that an asymptotic approximation (by Proposition 4.10) to this mean is used in Corollary 4.15.
Lemma 4.18. Consider the random variables defined in (4.1), and let where Proof. The first assertion is clear. For the second assertion let B   (a, b). (4.14) Here, the vectors a, b, f , the function v, and the matrix Σ are given as a = (a 1 , . . . , a m ) = lim Proof. This is simply a re-formulation of the results obtained in Proof Hence the joint distribution is given by with the constraints that m i=0 k i = N and n i=1 ik i = s N . Since there are two linearly independent linear constraints on the values of B (N ) we expect a (m − 1)-dimensional limiting distribution. Let x 0 , . . . , x m be real numbers with m i=0 x i = 0 and m i=0 ix i = 0, and let k i = N i+1 + √ N x i . By Stirling's approximation for the factorials for the numerator and the local limit theorem for lattice distributions for the denominator we see A check that the expression on the right-hand side is (considered as a function of x 2 , . . . , x m , say) the marginal density of (N (0, Σ)) 2,...,m with Σ as in4.15 concludes the proof.
For the convergence of moments we have here Proof. Again we restrict the exposition to the one component case and use the same notation as in the proof of Theorem 4.21. From the generating function given there we get P(S N = s N ) (4.18)

By the local central limit theorem for lattice distributions [17, Corollary VIII.3] we have
Applying this to the numerator and denominator shows that the quotients q r (N ) := are asymptotically of the form . Now, the asympotic assertion about the expectation follows immediately from the local limit theorem, applied to numerator and denominator in 4.16. The asymptotic assertion about the variance/covariance follows from the formulae above using the asymptotic form of q 1 (N ), q 2 (N ). Concerning the asserted convergence of the central fourth moment, note that the r − th factorial moment of B After expressing the central fourth moment as a linear combination of factorial moments and plugging in the asymptotical expressions for the q r (N ), a little algebra yields Remark 4.23. Let 1 N L (N ) −→ a = 0. A comparison to the unrestricted case, discussed in §4.3, shows that asymptotically the underlying total occupancy distributions are quite similar. They concentrate around the same expectations. In the unrestricted case the components of the limiting distribution 1 The components stay normal in the central restricted case, but the restriction causes an additional negative correlation between the components. This in turn forces the elements of the asymptotic covariance Σ of J (N ) to be smaller than in the unrestricted case, we compute Since J (N ) is a linear image of B (N ) its distribution is also asymptotically normal and it is clear from Theorem 4.21 and Corollary 4.15 that T (N ) is asymptotically normal and the preceding results show that the expectation resp. variance of T (N ) are of N 2 resp. N 3 , but the variance in the restricted case will (on the N 3 scale) be smaller than in the unrestricted case.
Again, let us emphasize the implications for fusion modules of the current algebra sl 2 ⊗ C[t].
and asymptotic variance 1 where σ 2 (a) = 1 12 m k=1 k(k + 2)a k , and c(i) = m k=i k+1−i k+1 a k . Proof. This is again a re-formulation of Theorem 4.21. Since J (N ) is a linear image of B (N ) , the asserted convergences follow from Proposition 4.10, Lemma 4.16, Proposition 4.22, and Remark 4.23. Note that by Proposition 4.10 the leading term of the expectation values, i.e. the coefficient of N 2 , depend only on a and b in the same manner as in the unrestricted case. Therefore, to derive the asymptotic mean given in (4.19) one simply has to replace the L k in the quadratic terms of (4.13) by their limit values a k .  N ) to the corresponding unrestricted one. We rewrite and let in the sequel It is straightforward to compute that in the case of jointly normal variables J  ) here under the simplifying assumption that the occupancy distribution is normal. Note that we know from Proposition 4.10 and Theorem 4.21 that the leading terms in N are correct.
(2) Comparing these results to (4.10) given at the end of the previous section, we see that the additional restriction has only small effects on the asymptotic distribution of T (N ) . Expectation and shape remain essentially unaltered, only the variance has decreased. This was to be anticipated from the probabilistic setup: the conditioning is "in line" with the law of large numbers.
(  N ). In this case the distribution of Y (N ) has been investigated under the heading "generalized Galois numbers" by Bliem and Kousidis [5] and later by Janson [22]. These authors studied the random variables Y (N ) with probability generating function where the constituents on the right hand side are independent, and the matrix Σ is given by ). Let us compare these results to our findings above. We obtain from Lemma 4. 18: which agrees with the expectation given in Theorem 4.27. Further, we have Finally, c = 0, where c = c(a, b) is as in Corollary 4.14. Thus, by Corollary 4.14 we have which is equivalent to the weak convergence assertion in Theorem 4.27, and hence establishes an independent proof. Moreover, by Corollary 4.14 that together with Theorem 4.1 independently proves Janson's Theorem 4.28.
Note that Theorem 4.28 as it stands does not generalize to more general distributions. As an example let B (N ) be multinomial with parameters N, p where p is not uniform. Here we get from Corollary 4.14 that The corresponding joint limiting distribution (on the right hand side of (4.23)) is normal, but the constituents are not independent.
For K = 1 this simplifies to Let us compare this to the only known result for a two component case in the literature, the case of Demazure modules V w (Λ) associated to the affine Kac-Moody algebra sl 2 . Fix the highest weight Λ = mΛ 0 + nΛ 1 . The random variables X w having probability generating function the basic specialization of the character χ(V w (Λ)) are given due to We can re-establish (4.26) by the computation of the left-hand side through the linearity of E(.) and the mean of the random variables S L , T L .
Since the results in [7] are derived differently and independently from our arguments through the analysis of Demazure's recursive character formula, this nicely confirms both, the correctness of the theory given here and of [7, . Fix a dominant integral weight Λ and a sequence (w (k) ) in W aff such that l(w (k) ) → ∞. Let µ (k) be the joint distribution of the degree and the finite weight in V w (k) (Λ). Letμ (k) be the distribution obtained from µ (k) by normalizing to a probability distribution and rescaling the two coordinates individually so that supp(μ (k) ) just fits into the rectangle [0, 1] × [−1, 1]. Then, as k → ∞, We consider only the Demazure modules V (s 1 s 0 ) N (mΛ 0 + nΛ 1 ) as the other cases can be derived equivalently. Let d N denote the maximal degree in these Demazure modules, i.e. d N = N 2 m+N (N −1)n. Let u = m+n = c, Λ denote the level of the representation, and consider the random variable with probability generating function given by the basic specialization of our Demazure module, that is The probability distribution of X N and 1 d N X N is the first coordinate of µ (N ) andμ (N ) for the Weyl group element w (N ) = (s 1 s 0 ) N , respectively. Now, equivalent to the asserted weak convergence ofμ (N ) we have and by (4.3) and Corollary 4.15 we have the convergences in distribution Since it is already known that the second coordinate ofμ (N ) concentrates in 0 by the factorization phenomenon (3.10), the claim follows.
The basic specialization can be interpreted as the generating function of dimensions of sl 2 representations in V w (Λ) splitted by the degree, i.e. the coefficient k in e −kα 0 . Therefore, Lemma 4.30 depicts their asymptotic concentration along the degree as a rational expression in the level of the Demazure module.
One can easily compute that the asymptotic concentration in the central restricted two component case (i.e. along a central string function) is the same. This exhibits the dominating weights in Demazure modules. Note that this is not deducible from [6, 7] due to the limitations of their arguments imposed by the non-positivity of Demazure's character formula. The basic argument that the concentrations coincide is that the expectation values of the random variables in the central restricted two component case have the same leading terms (in N ) as the unrestricted ones.

Local central limit theorems
It should be possible to prove the following local central limit theorem along the lines of [22, §6]. We pose them here as conjectures. . Then, uniformly in k as N → ∞, √ 2πσ N · P(X F N = k) = e −(k−µ N )/2σ 2 N + o(1). (5.1) Here, σ 2 N can be replaced by the explicit expression N 3 ( 1 4 f Σf t + v(a, b)) from (4.20). In particular, the dimension of the sl 2 submodule in F N of degree k grows as (5.1).
The analogue for the central restricted case is:  Then, uniformly in k as N → ∞, √ 2πσ · P(S N = k) = e −(k−µ)/2σ 2 + o(1) (5.2) In particular, the dimension of the h ⊂ sl 2 weight space with coordinates Note that those statements are the first hints towards an asymptotic description of the dimensions of fundamental submodules in fusion products available in the literature.
Then, we have a fermionic formula (a positive sum of products of qbinomial coefficients) for the graded character of the above fusion product F µ . That is, Proposition 6.2. Let µ = (µ 1 , . . . , µ m ) be a partition of M . Then, Proof. Let m ξ denote the monomial symmetric functions. Then, with K η,µ (q) = q n(µ) K η,µ (q −1 ) where n(µ) = i (i − 1)µ i as in [24, (3.10)] one has [24, Corollary 7.6]: Note that all partitions except µ have at most r entries, corresponding to the rank of the Lie algebra. Kirillov [25,26] is a great source of various combinatorial, geometric and statistical interpretations of q-supernomials S ξ,µ (q). Let us shortly remark on the geometric one.
Remark 6.4 (Cf. [5,22,27,43]). As pointed out by Kirillov [25,§1.4] it has been proven by Shimomura [37] that the q-supernomials count the number of rational points Fl µ ξ (F q ) over the finite field F q of the unipotent partial flag variety Fl µ ξ . To be precise, for a composition ξ ∈ Z r + of n, a ξ-flag in a n-dimensional vector space V is a sequence V 1 ⊂ · · · ⊂ V r such that dim V i = ξ 1 + · · · + ξ i . The set of all such flags is the partial flag variety Fl ξ . We let Fl µ ξ ⊂ Fl ξ be the subset of the partial flag variety Fl ξ consisting of the set of all ξ-flags F ∈ Fl ξ fixed by a unipotent endomorphism u ∈ Gl(V ) of type µ (a partition of n that describes the Jordan canonical form of u). Then, Fl µ ξ is a closed subvariety of Fl ξ , the so-called unipotent partial flag variety. Now, Shimomura [37] proves that the q-supernomials count the number of F q -rational points in Fl µ ξ . That is, with n(µ) as in (6.6) one has #Fl µ ξ (F q ) = q n(µ) S ξ,µ (q −1 ) = S * ξ,µ (q). (6.7)